Do you need to buy correct running shoes so that you do not injure yourself whilst running?

Answer 1

Yes.

You should figure out what your gait is (neutral, pronate, supinate). If you are the latter two it matters because your feet are not be properly absorbing shock, and the impact from running could lead to injuries later.

If you over-pronate your shoes will show uneven wear on the inside of the sole. Under-pronators (supinators) show wear on the outside. A neutral gait will show even wear. Sometimes the intensity of your arch coincides. If you can go to a running specialty store, they will film your feet on a treadmill, etc,

All the major/popular brands of shoes (Asics, Nike, Brooks, etc) make shoes for every kind of gait. You can often determine the right shoe with guides on those brands' websites.